What do Bollinger Bands actually measure?

Bollinger Bands, developed by John Bollinger in the early 1980s, look like a price channel. They are really a volatility measure wearing a price costume, and reading them any other way leads straight into the same trap as "overbought".

The construction

• Middle band = SMA(20) of closes
• σ = the population standard deviation of those same 20 closes — sum the squared distances from the average and divide by 20, not by 19
• Upper band = middle + 2σ · Lower band = middle − 2σ

Standard deviation is just the typical distance of those closes from their own average, so the bands are the 20-day average plus and minus twice that typical distance.

That "divide by 20" is worth pausing on, because it is the one detail that will stop your own calculation from matching a chart. Charting platforms use the population form. Most defaults you would reach for use the sample form and divide by 19 instead — Excel's STDEV and STDEV.S, R's sd, and the standard deviation of a pandas Series, whose default is the sample version. The sample form is larger by a fixed factor, the square root of 20 ÷ 19 = 1.0260, so it draws bands about 2.6% wider than the platform's, on every bar, in the same direction. Too small to look like a bug, too large to reconcile.

A worked example

The 20-day average is 100.0 and the standard deviation of those closes is 2.5:

• Upper band = 100 + 5 = 105
• Lower band = 100 − 5 = 95
• Band width = 10 points, or 10% of price

Now suppose the next fortnight is calm and σ falls to 1.0. The bands narrow to 102 and 98 — a width of 4%. Nothing changed about direction. Only dispersion changed. The bands moved because the market got quieter, which is precisely what they are built to show.

The squeeze

Narrow bands mark a quiet stretch. Traders watch squeezes because volatility clusters — calm periods tend to follow calm periods and violent ones follow violent ones, an observed statistical property of nearly every market. What clustering does not provide is a timer or a direction. "The bands are tight, so it must break soon, and upward" contains one observation and two inventions.

Clustering, drawn: every bar in the first half of this figure has a smaller range than every bar in the second.

The ±2σ fallacy

In a textbook normal distribution about 95% of observations fall within two standard deviations, which invites the conclusion that a close outside the bands is a once-a-month rarity. Three problems:

1. The twenty closes in the window are not twenty independent draws from a normal distribution — they are one serially correlated path, and the returns that generated them have fat tails. Extremes therefore land outside the bands more often than the bell curve allows.
2. σ is estimated from the same 20 bars, so the yardstick shifts as the thing being measured shifts.
3. Volatility clusters, so the excursions arrive in bunches rather than sprinkled evenly.

In a strong trend, price can walk along the upper band for weeks — the exact structural echo of RSI staying above 70. Bollinger's own long-standing caution is that a tag of a band is not a signal.

The literal reading

A close on the upper band means: today's close is two standard deviations above its own 20-day average — an unusually fast move by this instrument's own recent standards. That sentence is complete. Anything appended to it about tomorrow came from somewhere else.

In the data

Here are Apple's 20-day bands for the latest session, and its 20-day simple average for the same day:

The middle band and the average are the same number. These bands sit at the conventional two standard deviations. A chart drawn at 2.5 or at 1.5 is a different chart, and its band touches cannot be compared with these, so check the multiplier before you compare anyone's count of closes outside the bands.

Try it now

1. On the schematic above, find the widest bar of the quiet half and the narrowest bar of the violent half, and confirm the first is smaller. That is the clustering claim in full, and it says nothing whatever about when the second half begins.
2. Now measure real bands over the year below. First take the upper and lower band from the table in the section above, subtract, and express the width as a percentage of the middle band. Then count the closes that landed outside the upper or lower band over the last six months of the chart. Six months is about 125 sessions, so the textbook 5% is about six closes. Is your count near that — and were they scattered, or arriving in bunches?

3. Restate one of those closes as a literal sentence: "the close was two standard deviations above its own 20-day average." Notice that the literal version quietly refuses to tell you what to do.